considerations can be made for both conformations of ethane. However, since the
geometry of the former corresponds to higher molecular energy, its contribution to
the experimental coupling constant is smaller than that attributable to the latter. The
three-dimensional perspective view clearly defines the H1–C1–C2–H4 coupling
pathway through the bonds as a sequence of up-spikes in the proximity of the
nuclei. The spike pattern indicates that, in the eclipsed conformation, through-space
H1–H4 coupling takes place to a little extent, see Fig. 7.45a, b, where down-spikes
connected with the current vortex about the H1–C1 bond are observed. These
down-spikes are absent in Fig. 7.45c, d for the staggered conformation, in which
through-space vicinal coupling is unimportant. Although the up-spikes about the
carbon nuclei are much lower than those over the coupled protons, they indicate the
essential role of the charge distribution in these regions for transporting spin
information.
The interpretation of the coupling density maps of Fig. 7.45 via the approaches
of Refs. [51, 52] is facilitated by models of the interference pattern of current
densities induced by the pair of interacting nuclear magnetic dipoles. Thus, for a
reference electron of given spin close to H1, the Fermi correlation precludes
same-spin polarization in its proximity, gives rise to alternating opposite spin
densities along the coupling pathway, and determines the indirect spin-spin interaction for two nuclei at its ends.
7.9 Conclusions
The response of diamagnetic molecules to an external homogeneous static magnetic
field B and to intramolecular permanent nuclear magnetic dipoles m I is effectively
rationalized via maps of streamlines and modulus of quantum mechanical induced
current densities J
B and J
m I . In this chapter it is shown that the essential features of
intrinsic tensor properties, magnetizability v ab , magnetic shielding r
I
ab at nucleus I,
and spin-spin coupling K
I a J b between nuclei I and J, are nicely explained by analyzing contributions which arise from different molecular domains. In particular,
r
I
ab and K
I a J b can be related to property-density functions R
I
ab and j
I a J b via noninvertible maps, f : J
B
ðrÞ ! R
I
ðrÞ and f : J
m I ðrÞ ! j
IJ
ðrÞ, which clearly visualize
the local phenomenology. The topological analysis of singularities, stagnation lines
and stagnation graph of J
B and J
m I current density vector fields provides powerful
interpretative tools, as shown in a number of examples. A definition of
magnetic-field induced delocalized electron currents (ring currents) can be proposed
on a topological criterion.
222
P. Lazzeretti
geometry of the former corresponds to higher molecular energy, its contribution to
the experimental coupling constant is smaller than that attributable to the latter. The
three-dimensional perspective view clearly defines the H1–C1–C2–H4 coupling
pathway through the bonds as a sequence of up-spikes in the proximity of the
nuclei. The spike pattern indicates that, in the eclipsed conformation, through-space
H1–H4 coupling takes place to a little extent, see Fig. 7.45a, b, where down-spikes
connected with the current vortex about the H1–C1 bond are observed. These
down-spikes are absent in Fig. 7.45c, d for the staggered conformation, in which
through-space vicinal coupling is unimportant. Although the up-spikes about the
carbon nuclei are much lower than those over the coupled protons, they indicate the
essential role of the charge distribution in these regions for transporting spin
information.
The interpretation of the coupling density maps of Fig. 7.45 via the approaches
of Refs. [51, 52] is facilitated by models of the interference pattern of current
densities induced by the pair of interacting nuclear magnetic dipoles. Thus, for a
reference electron of given spin close to H1, the Fermi correlation precludes
same-spin polarization in its proximity, gives rise to alternating opposite spin
densities along the coupling pathway, and determines the indirect spin-spin interaction for two nuclei at its ends.
7.9 Conclusions
The response of diamagnetic molecules to an external homogeneous static magnetic
field B and to intramolecular permanent nuclear magnetic dipoles m I is effectively
rationalized via maps of streamlines and modulus of quantum mechanical induced
current densities J
B and J
m I . In this chapter it is shown that the essential features of
intrinsic tensor properties, magnetizability v ab , magnetic shielding r
I
ab at nucleus I,
and spin-spin coupling K
I a J b between nuclei I and J, are nicely explained by analyzing contributions which arise from different molecular domains. In particular,
r
I
ab and K
I a J b can be related to property-density functions R
I
ab and j
I a J b via noninvertible maps, f : J
B
ðrÞ ! R
I
ðrÞ and f : J
m I ðrÞ ! j
IJ
ðrÞ, which clearly visualize
the local phenomenology. The topological analysis of singularities, stagnation lines
and stagnation graph of J
B and J
m I current density vector fields provides powerful
interpretative tools, as shown in a number of examples. A definition of
magnetic-field induced delocalized electron currents (ring currents) can be proposed
on a topological criterion.
222
P. Lazzeretti